非齐次Riemann-Liouville分数阶微分方程解的性质

1、非齐次Riemann-Liouville分数阶微分方程解的性质研究摘要分数阶微积分是数学分析里面的一个重要分支，其理论主要是研究和探讨非整数 阶微积分的数学性质及其相关应用，是整数阶微积分的延伸和拓展。近年来，分数阶 微积分的应用覆盖多种领域，其中包括材料和力学系统、流变学、信号处理和系统辨 识以及刻画具有记忆和遗传性质的各种物质等重要领域，因此现实世界对分数阶微积 分的需求也越来越强烈，本文主要针对分数阶微积分中的非齐次Riemann- Liouville分数阶微分方程解性质展开讨论和研究。本文的主要工作有：第一，根据Riemann- Liouville分数阶微积分的定义，对其基本性质做了一

2、些探讨， 紧接着给出Riemann-Liouville分数阶微分方程定义，为了更好的去研究非齐次Riema nn - Liouville方程的性质，对于齐次 Riema nn -Liouville分数阶微分方程的研究 也是必要的。第二，利用微分与积分的转化以及 Laplace变换推导出非齐次Riemann- Liouville 方程解的基本表达形式，根据其强解和温和解的定义对非齐次Riemann- Liouville方程解的基本性质做出具体的研究。第三，根据非齐次Riemann-Liouville方程强解和温和解的定义对其存在性和唯一 性进行讨论和证明。关键词：分数阶；非齐次；强解；温和解；存

3、在唯一性IABSTRACTFractional calculus is an important branch of mathematical analysis, its theory is mainly to study and discuss the mathematical properties and related applicati ons of non in teger calculus, and it is the exte nsion and expa nsion of in teger calculus. In rece nt years, the fractional c

4、alculus has been widely used in different areas, these areas are mostly including the materials and mecha ni cal systems, rheology, sig nal process ing and system ide ntificati on and characterizatio n of memory and hereditary properties of various substa nces eAt. the same time, the dema nd of the

5、fractio nal calculus in the real world is more and more in ten se, so discussi on and research in this paper mainly for the fracti onal calculus of non homoge neous Riema nn-Liouville fractio nal differe ntial equati on soluti on properties.The main work of this paper is as follows:Firstly, accordin

6、g to the definition of Riemann-Liouville fractional calculus, its basic properties are discussed, and the n gives the Riema nn-Liouville fractio nal differe ntial equation is defined, in order to study the non homogeneous Riemann-Liouville equation, is n ecessary for the study of homoge neous Riema

7、nn-Liouville fractional differe ntial equati ons.Secondly, the use of differential and integral transformation and Laplace transformation are non homoge neous Riema nn-Liouville equati ons accordi ng to the basic form of expressi on, strong solution and mild solutions defined on the specific study o

8、f non basic properties of homoge neous Riema nn-Liouville equati on to make.Thirdly, the existence and uniqueness of the strong solution and the mild solution of the non homoge neous Riema nn-Liouville equati on are discussed and proved.KEY WORDS : Fractional order; Nonhomogeneous; Strong solution;

9、Mild solution; Existence and Uniquen essII1绪论11.1研究背景及意义11.1.1 研究背景11.1.2研究意义11.2研究现状21.3主要工作32预备知识52.1分数阶微积分52.1.1 Riemann-Liouville分数阶微积分的定义52.1.2 Caputo分数阶导数的定义 52.1.3 Riemann-Liouville分数阶导数与 Caputo分数阶导数的对比 62.1.4 Riemann-Liouville分数阶微分方程的定义 62.2 Riemann-Liouville 分数阶预解式 72.3 Riemann-Liouville 分数阶余弦函数 92.4相关定义及定理113 ( FAQ x,y